Mathematical and physical meaning of the crossings of energy levels in ${\cal PT}-$symmetric systems

TL;DR

This paper explores the nature of energy level crossings in ${ m PT}$-symmetric systems, revealing how passing through exceptional points can cause significant physical changes without energy complexification, supported by solvable models.

Contribution

It provides a new interpretation of energy level crossings in ${ m PT}$-symmetric systems, linking them to exceptional points and phase transitions.

Findings

Energy crossings can lead to dramatic physical changes at exceptional points.

Passing through an EP can cause phase transitions without energy complexification.

Exactly solvable models illustrate the theoretical findings.

Abstract

Unavoided crossings of the energy levels due to a variation of a real parameter are studied. It is found that after the quantum system in question passes through one of its energy-crossing points {\it alias} Kato's exceptional points (EP), its physical interpretation may {\em dramatically} change even when the crossing energies themselves do not complexify. The anomalous physical phase-transition mechanism of the change is revealed, attributed to the EP-related mathematics and illustrated via several exactly solvable matrix toy models.